#151317

order	512
nilpotency class	5
isoclinism	isoclinic class of #151316
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12181
commutator, Frattini, cosocle	#423	C₂³
prime radical	C₂³	#454
LCS	G ▷ #423 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #423 ◁ G
LECS	G ▷ #423 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

198 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,8,14)(3,15,19,46)(4,20,22,10)(5,25,28,43)(6,13,7,12)(9,32,35,17)(11,31,41,29)(16,47,51,61)(18,39,30,45)(21,56,57,40)(23,37,24,38)(26,34,53,42)(27,55,58,36)(33,48,54,59)(44,50,64,49)(52,62,60,63);(1,3,7,18,8,19,6,30)(2,9,13,34,14,35,12,42)(4,16,24,50,22,51,23,49)(5,26,31,17,28,53,29,32)(10,33,38,63,20,54,37,62)(11,39,43,15,41,45,25,46)(21,52,58,48,57,60,27,59)(36,44,56,61,55,64,40,47);(1,4,7,24,8,22,6,23)(2,10,13,38,14,20,12,37)(3,16,18,50,19,51,30,49)(5,27,29,57,28,58,31,21)(9,33,34,63,35,54,42,62)(11,40,25,55,41,56,43,36)(15,44,39,47,46,64,45,61)(17,52,26,59,32,60,53,48)

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=g,d^2=fi,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=ehi,f^a=fi,g^a=gh,d^b=dfi,e^b=eghi,f^b=fi,h^b=hi,d^c=dg,e^c=ei,e^d=eh,g^d=gi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58513);